The relation $$R = \{(a, b) : gcd(a, b) = 1, 2a \neq b, a, b \in \mathbb{Z}\}$$ is:

We need to determine the properties of $$R = \{(a, b) : \gcd(a, b) = 1, 2a \neq b, a, b \in \mathbb{Z}\}$$.

Reflexivity: For $$(a, a) \in R$$: need $$\gcd(a, a) = 1$$, which requires $$|a| = 1$$. Since $$(2, 2) \notin R$$, R is not reflexive.

Symmetry: Consider $$(2, 1)$$: $$\gcd(2, 1) = 1$$ and $$4 \neq 1$$, so $$(2, 1) \in R$$. But $$(1, 2)$$: $$\gcd(1, 2) = 1$$ and $$2(1) = 2 = b$$, so $$(1, 2) \notin R$$. R is not symmetric.

Transitivity: Consider $$a = 4, b = 3, c = 8$$. $$(4,3) \in R$$ since $$\gcd(4,3)=1$$ and $$8 \neq 3$$. $$(3,8) \in R$$ since $$\gcd(3,8)=1$$ and $$6 \neq 8$$. But $$(4,8) \notin R$$ since $$\gcd(4,8)=4 \neq 1$$. R is not transitive.

The correct answer is Option 4: neither symmetric nor transitive.